# The Ellipse

Viewing the conic section

maps out on the DC sector plane, we know that it follows an elliptical path centered on it's x axis. Performing a passive rotation on points in the DC section plane does not physically change the position in space, i.e. passive rotations only give the components in a new coordinate system. Once such a rotation has been performed, the equation describing these points must be done within that plane.An ellipse centered at the origin can be expressed in the form

For an ellipse not centered on the origin, but instead the point (h',k'), this expression becomes

In the plane of the DC sector, this equation becomes

where the center of the ellipse is found at

.

Switching to the frame of the wires, the ellipse is still centered at in the DC sector, with the semi-major axis lying on the x' axis. For a rotation in the y-x plane, this corresponds to a positive angle \[Theta], with the rotation matrix R(Subscript[\[Theta], yx]). In the frame of the wires, this center point falls at

(x
y
z

)=(cos (6\[Degree]) -sin (6\[Degree]) 0 sin (6\[Degree]) cos (6\[Degree]) 0 0 0 1

) . (x' y' z'

)

(x y z

)=(cos (6\[Degree]) -sin (6\[Degree]) 0 sin (6\[Degree]) cos (6\[Degree]) 0 0 0 1

) . (\[CapitalDelta]a 0 0

)

(x y z

)=(\[CapitalDelta]a cos (6\[Degree]) \[CapitalDelta]a sin (6\[Degree]) 0

)

-> (x ", y*, z*Subscript[), Center] = (\[CapitalDelta]a cos 6\[Degree], \[CapitalDelta]a sin 6\[Degree], 0) = (h*, k*, 0)

Performing an active rotation, we will rotate the equation for an ellipse in the frame of the DC to the frame of the wires . In the frame of the DC, the ellipse is centered on the x' axis, with the intersection points not having a uniform spacing in the ellipse parameter. In the frame of the wires, the ellipse is tilted 6\[Degree] counterclockwise from the xaxis, with the intersection points having a uniform spacing in the ellipse parameter.

(x
y
z

)=(cos 6\[Degree] sin 6\[Degree] 0 -sin 6\[Degree] cos 6\[Degree] 0 0 0 1

) . (x' y' z'

)

(x
y
z

)=(x'cos 6\[Degree]+y' sin 6\[Degree] -x'sin 6\[Degree]+y'cos 6\[Degree] 0

)

Substituting this into the equation for an ellipse in the frame of the wires,

(x*+h*)^2/a^2+(y*+k*)^2/b^2=1

((x'+\[CapitalDelta]a Cos[6 \[Degree]])Cos[6 \[Degree]]+(y'+\[CapitalDelta]a Sin[6 \[Degree]])Sin[6 \[Degree]])^2/a^2+(-(x'+\[CapitalDelta]a Cos[6 \[Degree]])Sin[6 \[Degree]]+(y'+\[CapitalDelta]a Sin[6 \[Degree]])Cos[6 \[Degree]])^2/b^2=1

(x'Cos[6 \[Degree]]+\[CapitalDelta]a (Cos^2)[6 \[Degree]]+y'Sin[6 \[Degree]]+\[CapitalDelta]a (Sin^2)[6 \[Degree]])^2/a^2+(-x'Sin[6 \[Degree]]-\[CapitalDelta]a Cos[6 \[Degree]]Sin[6 \[Degree]]+y'Cos[6 \[Degree]]+\[CapitalDelta]a Sin[6 \[Degree]]Cos[6 \[Degree]])^2/b^2=1

(x'Cos[6 \[Degree]]+y'Sin[6 \[Degree]]+\[CapitalDelta]a (Cos^2)[6 \[Degree]]+\[CapitalDelta]a (Sin^2)[6 \[Degree]])^2/a^2+(-x'Sin[6 \[Degree]]+y'Cos[6 \[Degree]]+\[CapitalDelta]a Cos[6 \[Degree]]Sin[6 \[Degree]]-\[CapitalDelta]a Cos[6 \[Degree]]Sin[6 \[Degree]])^2/b^2=1

(x'Cos[6 \[Degree]]+y'Sin[6 \[Degree]]+\[CapitalDelta]a ((Cos^2)[6 \[Degree]]+ (Sin^2)[6 \[Degree]]))^2/a^2+(-x'Sin[6 \[Degree]]+y'Cos[6 \[Degree]]+\[CapitalDelta]a (Cos[6 \[Degree]]Sin[6 \[Degree]]-Cos[6 \[Degree]]Sin[6 \[Degree]]))^2/b^2=1

(x'Cos[6 \[Degree]]+y'Sin[6 \[Degree]]+\[CapitalDelta]a )^2/a^2+(-x'Sin[6 \[Degree]]+y'Cos[6 \[Degree]])^2/b^2=1